Fsc Part 1 Mathematics (Complete Solution)

# Q14 Verify that

Verify that (AB)-1=B-1A-1 if

(i)  A =

 1 2 -1 0

,  B=

 -3 1 4 -1

SOLUTION: Now taking L.H.S

AB=

 1 2 -1 0

 -3 1 4 -1

=

 -3+8 1-2 3+0 -1+0

=

 5 -1 3 -1

lABl=

 5 -1 3 -1

=-5+3

=-2

(AB)-1=
=
 1 -2

 -1 1 -3 5

=
 1 2

 1 -1 3 -5

=

 1/2 -1/2 3/2 -5/2

.........(i)

Taking R.H.S

lAl=

 1 2 -1 -0

=0+2

=2

A-1=
=
 1 2

 0 -2 1 1

 0 -1 1/2 1/2

lBl=

 -1 1 4 -3

=3-4

=-1

B-1=
=
 1 -1

 -1 -1 -4 -3

 1 1 4 3

B-1A-1=

 1 1 4 3

 0 -1 1/2 1/2

=

 (0+1)/2 (-2+1)/2 (0+3)/2 (-8+3)/2

=

 1/2 -1/2 3/2 -5/2

.........(ii)

(i)=(ii)

L.H.S=R.H.S

Hence (AB)-1=B-1A-1

SO PROVED

(ii)  A =

 5 1 2 2

,  B=

 4 3 2 1

SOLUTION: Now taking L.H.S

AB=

 5 1 2 2

 4 3 2 1

=

 20+2 15+1 8+4 6+2

=

 22 16 10 8

lABl=

 22 16 12 8

= 176-192

= -16

(AB)-1=
=
 1 -16

 8 -16 -12 22

=
 1 16

 -8 16 12 -22

=

 -1/2 1 3/4 -11/8

.........(i)

Taking R.H.S

lAl=

 5 1 2 2

=10-2

=8

A-1=
=
 1 8

 2 -1 -2 5

 1/4 -1/8 -1/4 5/8

lBl=

 4 3 2 1

=4-6

=-2

B-1=
=
 1 -2

 1 -3 -2 4

 -1/2 3/2 1 -2

B-1A-1=

 -1/2 3/2 1 -2

 1/4 -1/8 -1/4 5/8

=

 (-2-6)/16 (1+15)/16 (4+8)/16 (-2-20)/16

=

 -1/2 1 3/4 -11/8

.........(ii)

(i)=(ii)

L.H.S=R.H.S

Hence (AB)-1=B-1A-1

SO PROVED